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2022. Т. 40. С. 104—111
Онлайн-доступ к журналу: http://mathizv.isu.ru
Серия «Математика»
Research article УДК 510.67
MSC 03C30, 03C15, 03C50, 54A05
DOI https://doi.org/10.26516/1997-7670.2022.40.104
Approximations of Acyclic Graphs
Nurlan D. Markhabatov
Novosibirsk State Technical University, Novosibirsk, Russian Federation И markhabatov@gmail.com
Abstract. In this paper, approximations of acyclic graphs are studied. It is proved that any theory of an acyclic graph (tree) of finite diameter is pseudofinite with respect to acyclic graphs (trees), that is, any such theory is approximated by theories of finite structures (acyclic graphs, trees). It is also proved that an acyclic graph of infinite diameter with infinite number of rays is pseudofinite.
Keywords: approximation of theory, tree, acyclic graph, pseudofinite theory
Acknowledgements: The author expresses his deep gratitude to his supervisor Sergei Vladimirovich Sudoplatov for an interesting research topic and valuable comments. The author is also grateful to the referee for a few clarifying remarks and for the patience with the author's poor grammar.
The research was financially supported by Committee of Science in Education and Science Ministry of the Republic of Kazakhstan (Project No. AP08855497), Russian Foundation for Basic Researches (Project No. 20-31-90003).
For citation: Markhabatov N. D. Approximations of Acyclic Graphs. The Bulletin of Irkutsk State University. Series Mathematics, 2022, vol. 40, pp. 104-111. https://doi.org/10.26516/1997-7670.2022.40.104
Научная статья
Аппроксимации ациклических графов Н. Д. Мархабатов
Новосибирский государственный технический университет, Новосибирск, Российская Федерация И markhabatov@gmail.com
Аннотация. В статье исследуются аппроксимации теории ациклических графов. Доказано, что теория любого ациклического графа (дерева) конечного диаметра
псевдоконечна, то есть любая такая теория аппроксимируется теориями конечных структур (ациклических графов, деревьев). Также доказано, что ациклический граф бесконечного диаметра с бесконечным числом лучей псевдоконечен.
Ключевые слова: аппроксимация теории, дерево, ациклический граф, псевдоконечная теория
Благодарности: Автор выражает глубокую благодарность своему научному руководителю Сергею Владимировичу Судоплатову за интересную тему для написания и ценные замечания. Также автор благодарен рецензенту за несколько уточняющих замечаний и за терпение по поводу плохой грамматики автора.
Работа выполнена при финансовой поддержке Комитета науки в сфере образования и науки РК (проект AP08855497), РФФИ (проект 20-31-90003).
Ссылка для цитирования: Markhabatov N. D. Approximations of Acyclic Graphs // Известия Иркутского государственного университета. Серия Математика. 2022. Т. 40. C. 104-111.
https://doi.org/10.26516/1997-7670.2022.40.104
1. Preliminaries
Throughout the article, we use the terminology from [1; 3; 8]. We also continue to study the approximations of various structures [4-7].
1.1. Preliminaries from graph theory
Recall the following basic concepts. Let L = {R}, where R is a binary relation symbol. A graph is a structure r = (G,R). The elements of the universe G are called the vertices of the graph r, and the elements of the binary relation R C G2 are edges. The edge connecting the vertices a and b is denoted by [a, b]. If a vertex a £ G is an endpoint of edge u £ R, then a and u are incident. The degree of vertex a in a graph r, written degr(a) or simply deg(a), is the number of edges incident to a, except that each loop at a counts twice. A vertex of degree 0 is called isolated, a vertex of degree 1 is called leaf or hanging vertex. A graph that contains no cycles is called an acyclic graph. A connected acyclic graph is called a tree. Any graph without cycles is also called a forest so that the components of a forest are trees. Subsystems r' = (G', R') of the graph r are called subgraphs. If r' C r and r' contains all the edges [x, y] £ R' with x,y £ G', then r' is an induced subgraph of r.
Theorem 1. [3] Any nontrivial finite tree has at least two hanging vertices.
Recall that a connected component of the graph r = (G, R) is a connected subgraph of the graph r that is maximal by inclusion. Any connected component C of the graph r is uniquely determined by any of its
elements a e G and is denoted by С(a,G) or by С(a,R), when it is clear what universe G go speech.
A route in the graph Г is any non-empty sequence
r(ao,an) = (ao, ...,an)
of vertices for which Г = R(ai,ai+1),i = 0,.. .,n. In this case, the route f(a0,an) will also be called (a0, an_i)-route (or n-route), and the natural number n is the length of the route r(a0, an).
A path is a simple graph whose vertices can be ordered so that two vertices are adjacent if and only if they are consecutive in the list. If for two vertices a,b e G there is a path connecting them, then there is sure to be a minimal path connecting these vertices. We denote the length of this path by p(a, b). If Г has no such path, then p(a, b) = те. A tree is a path if and only if deg(a) < 2 for all vertice a of the tree.
Definition 1. [8] For a fixed vertex a, the value e(a) ^ max{p(a, b)\b e G} is called the eccentricity of the vertex a. The eccentricity of the vertex is equal to the distance from this vertex to the most distant from it. The maximum among all the eccentricities of the vertices is called the diameter of the graph Г and is denoted by d(r) : d(r) ^ max{e(a)\a e G}. The vertex a is called peripheral if e(a) = ^(Г). The minimal eccentricity of the graph Г is called its radius and is denoted by г(Г) : г(Г) ^ min{e(a)\a e G}. The vertex a is called central if e(a) = г(Г). The set of all central vertices of a graph is called its center.
Definition 2. [2] An infinite graph Г = (G, R) of the form G = {a0, a1, a2,.. .},R = {(a0, a1), (a1,a2), (a2, a3),...} is called a ray, and a double ray is an infinite graph Г = (G, R) of the form G = {..., a_2, a_1,a0, a1,a2,...}, R = {..., (a_2,a_1), (a_1,a0), (a0,a1), (a1,a2),...}; in both cases the an are assumed to be distinct.
1.2. Preliminaries from model theory and approximations of theories
In 1968, James Ax, in his work [1], first introduced the concept of pseudofiniteness to show the decidability of the theory of all finite fields, i.e. there is an algorithm to decide whether a given statement is true for all finite fields. It was proved that pseudofinite fields are exactly those infinite fields that have every elementary property common to all finite fields, that is, pseudofinite fields are infinite models of the theory of finite fields. Pseudofinite structures after Ax have not been studied in general for a long time. Until the 1990s, only a few results were obtained on this topic. After a long break, pseudofinite structures began to be studied in the joint works of E. Hrushovski and G. Cherlin.
Definition 3. [1] An L-structure M is pseudofinite if for all L-sentences <p, M = <p implies that there is a finite Mo such that Mo = f. The theory T = Th(M) of the pseudofinite structure M is called pseudofinite.
In classical logic, the following property is a straightforward consequence of pseudofiniteness.
Proposition 1. [1] Let M be a pseudofinite structure and f : Mk ^ Mk be a definable function. Then f is injective if and only if f is surjective.
The structure of relational language L with universe |JieI Mi and interpretations of relation symbols from L represented as a combination of their interpretations in the structures Mi, i £ I, is called a disjoint union \_\iei Mi of disjoint structures Mi, i £ I.
Definition 4. [7] Let T be a family of theories and T be a theory such that T £ T. The theory T is said to be T-approximated, or approximated by the family T, or a pseudo-T-theory, if for any formula <p £ T there exists T' £ T for which ^ £ T'.
If the theory T is T-approximated, then T is said to be an approximating family for T, and theories T' £T are said to be approximations for T.
2. Classes of acyclic graphs (trees)
We denote the following classes of acyclic graphs and consider various approximations for these classes.
Let Qfin(X), for arbitrary cardinality X, be the family of all infinite acyclic graphs consisting of X connected components of bounded in aggregate diameters.
Acyclic graph from the class Qfin(1) is an tree, that is, it consists of a single connected component with infinitely many hanging vertices.
Acyclic graphs from the class Qfin(X), for arbitrary cardinality A, can contain both finite and infinite components of bounded diameters.
Let Qinf (A), for arbitrary cardinality X, be the family of all infinite acyclic graphs consisting of X connected components of infinite diameters.
It is clear that any component of the graph from Qinf (A) is infinite. An infinite component has an infinite diameter if it contains at least one ray. Acyclic graphs from the class Qinf (1) may contain a finite or infinite number of hanging vertices or not have hanging vertices. An example of an acyclic graph from this class with only one hanging vertex is ray. And an example of an acyclic graph from this class without hanging vertices is double ray.
Acyclic graphs from the class Ginf (2) consists of two connected components of infinite diameters. Each component can have an unbounded number of hanging vertices.
For theories of graphs of the above-defined classes, we will consider various approximations (approximation by theories of acyclic graphs or trees and cycles).
3. Approximations of acyclic graphs from the class Ginf (A)
Let us consider how the theories of an acyclic graphs from the class Ginf (A) with a finite number of rays are approximated by theories of finite structures. By the number of rays, we mean the number of rays emanating from a fixed number of vertices.
Theorem 2. Let T be the theory of an acyclic graph Г from the class Ginf (A), for finite cardinality X, with finitely many rays. If the number of rays in Г is even then T is pseudofinite theory.
Proof. Let Г = (G, R), G = ЦieX Gi, where Gi are trees. We fix a partition of the set of rays of the graph Г into pairs. For an arbitrary n £ ш and a tuple a = (a0,... ,an-i) £ G, where ai lies to different connected components of the graph Г, we define a finite graph Г„,га as follows: to the union of n-neighborhoods of vertices ai, 0 < i < n-1, we add the edges that connect the ends of pairs of rays belonging to the union of n-neighborhoods of vertices ai, 0 < i < n — 1. Then the theory T is approximated by the set [Тк(Гй,п)1п £ w,a = (ao,...,an-i) £ G, Vi(ai £ Gki ),Gki П Gkj = (i = j)}. □
Theorem 3 immediately implies the following:
Corollary 1. For any ^ > 1, there is a pseudofinite acyclic graph with ^ hanging vertices.
Theorem 3. The theory T of an infinite acyclic graph Г £ Ginf (X) for arbitrary cardinality X with an infinitely many rays is pseudofinite.
Proof. Let T be a theory of an acyclic graph Г with infinite number of rays. The set of theories of all subgraphs Г' of graph Г such that Г' £ Ginf (X) for some finite ^ < X, Г' with an even number of rays, approximates the theory T = ТЛ,(Г), so, by Theorem 2, T is pseudofinite. □
Example 1. We consider the theory T of an acyclic graph, which is a double ray. We denote the double ray by Г = (G,R), where G = {..., a-2, a-i, ao, ai,
a2,...}, R = {..., (a-2, a-i), (a-i,a0), (a0, ai), (ai, a2),...}. The thoery T
can be approximated by a set of theories of cycles of increasing length. So the theory T is pseudofinite.
Example 2. We consider the theory T of an infinite acyclic graph r from the class Qinf (A), consisting of A infinite components of infinite diameters without hanging vertices. It is clear that components of r are double rays. As in Example 1, the theory T can be approximated by a set of theories of cycles of increasing length. So the theory T is pseudofinite.
4. Approximations of acyclic graphs from the class Qfin(A)
The following theorem shows that any theory of an acyclic graph from the class Gfm(X) is approximated by theories of finite structures.
Theorem 4. Theory T of any infinite acyclic graph r from the class Qfin(X), for arbitrary cardinality X, is pseudofinite.
Proof. Let the theory T have an infinite model r = (G; R) from the class Gfin(^). Let r = To U r with universes Go, G1 respectively, where To is a subgraph consisting of k finite components of bounded diameters, for some cardinality k, and r is a subgraph consisting of p infinite components of bounded diameters, for some cardinality p.
Let r0 be the disjoint union of trees T^, i e k. Then the theory Th(r0) is approximated by the set [Th(Y[r^) | I C k, |/1 < w}.
Let r1 be the disjoint union of trees r'[,i e k, and ai be the vertices from the centers of the trees r". Then the theory Th(r1) is approximated by the set [Th(UieI T") | I C k, |J| < u, rf is the finite subgraph of r'[ and ai is the vertex from the center of T"}. □
As the following theorem shows, a theory of acyclic graphs from the class Qfin(X), A > 2, does not approximated by theories of finite trees.
Theorem 5. Let r be an acyclic graph from class Qfin(X), for a cardinality X. The theory T = Th(r) is approximated by theories of finite trees if and only if X = 1.
Proof. Let T be the theory of an acyclic graph r = (G; R) consisting of A connected components of bounded diameters and suppose that T is approximated by theories of finite trees. Suppose d is the maximum diameter of the connected components of the graph r and let A > 1. Consider the sentence the distance between any two vertices is either greater than d +1 or less than or equal to d; there are two vertices, the distance between which is greater than d + 1. It is clear that the sentence $ is true in r but false in any tree, which contradicts the approximability of Th(r) by the set of theories of trees.
Proved as in the proof of Theorem 4. □
5. Conclusion
The work investigates the approximations of theories of acyclic graphs of the classes Gfin(X), Qinf (A). It is proved that any acyclic graph from the class Gfin(X) is pseudofinite. It is also proved that an acyclic graph of infinite diameter with infinite number of rays is pseudofinite.
References
1. Ax J. The elementary theory of finite fields. Ann. Math., 1968, vol. 88, no. 2, pp. 239-271.
2. Diestel R. Graph theory. New York, Springer, Heidelberg, 2005. 422 p.
3. Harary F. Graph Theory. Addison-Wesley, 1969. 274 p.
4. Kulpeshov B.Sh., Sudoplatov S.V. Ranks and approximations for families of ordered theories. Algebra and Model Theory 12. Collection of papers. Eds. Pinns A.G., Poroshenko E.N, Sndoplatov S.V.. Novosibirsk, NSTU Publ.,2019, pp. 32-40.
5. Markhabatov N.D. Pseudofiniteness of locally free algebras. Algebra and Model Theory 12, Collection of papers. Eds. Pinns A.G., Poroshenko E.N., Sndoplatov S.V.. Novosibirsk, NSTU Publ., 2019, pp. 41-46.
6. Markhabatov N.D. Ranks for Families of Permutation Theories. The Bulletin of Irkutsk State University. Series Mathematics, 2019, vol. 28, pp. 85-94. https://doi.org/10.26516/1997-7670.2019.28.85
7. Sudoplatov S.V. Approximations of theories. Siberian Electronic Mathematical Reports, 2020, vol. 17, pp. 715-725. https://doi.org/10.33048/semi.2020.17.049
8. Sudoplatov S.V., Ovchinnikova E.V. Diskretnaya matematika [Discrete mathematics]. Moscow, Urait Publ., 2022, 280 p. (in Russian)
Список источников
1. Ax J. The elementary theory of finite fields // Ann. Math. 1968. Vol. 88, N 2. P. 239-271.
2. Diestel R. Graph theory. New York : Springer, Heidelberg, 2005. 422 p.
3. Harary F. Graph Theory. Addison-Wesley, 1969. 274 p.
4. Kulpeshov B. Sh., Sudoplatov S. V. Ranks and approximations for families of ordered theories // Algebra and Model Theory 12. Collection of papers / eds. A. G. Pinus, E. N. Poroshenko, S. V. Sudoplatov. Novosibirsk : NSTU, 2019. P. 32-40.
5. Markhabatov N. D. Pseudofiniteness of locally free algebras // Algebra and Model Theory 12, Collection of papers / eds. A. G. Pinus, E. N. Poroshenko, S. V. Sudoplatov. Novosibirsk : NSTU, 2019. P. 41-46.
6. Markhabatov N. D. Ranks for Families of Permutation Theories // Известия Иркутского государственного университета. Серия Математика. 2019. Т. 28. С. 85-94. https://doi.org/10.26516/1997-7670.2019.28.85
7. Sudoplatov S. V. Approximations of theories // Сибирские электронные математические известия. 2020. Т. 17. С. 715-725. https://doi.org/10.33048/semi.2020.17.049
8. Судоплатов С. В., Овчинникова Е. В. Дискретная математика : учебник и практикум. М. : Юрайт, 2022. 280 с.
Об авторах
Мархабатов Нурлан Дарханулы,
аспирант, ассистент, Новосибирский государственный технический университет, Российская Федерация, 630073, г. Новосибирск, markhabatov@gmail.com, https://orcid.org/0000-0002-5088-0208
About the authors Nurlan D. Markhabatov,
Postgraduate, Assistant, Novosibirsk State Technical University, Novosibirsk, 630073, Russian Federation, markhabatov@gmail.com, https://orcid.org/0000-0002-5088-0208
Поступила в 'редакцию / Received 02.04.2021 Поступила после рецензирования / Revised 26.02.2022 Принята к публикации / Accepted 01.03.2022
